Open Access
Issue
A&A
Volume 683, March 2024
Article Number A233
Number of page(s) 22
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/202348428
Published online 22 March 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

The variability of Y Ophiuchi (Y Oph), with a pulsation period of about 17 days, was discovered some 130 yr ago (Sawyer 1890, 1892; Luizet 1905). In the same epoch, the period-luminosity relation (hereafter, the PL relation) of Cepheids was likewise discovered (Leavitt 1908; Leavitt & Pickering 1912). However, Y Oph is generally not taken into account in the calibration of the PL relation because of its unusual behavior. Indeed, using the Baade-Wesselink (BW) method, Abt (1954) found that this star is 1 mag fainter than the result derived using the PL relation. In other words, the luminosity derived by the BW method corresponds to a Cepheid with a pulsation period of ten days or less.

A number of variations on the BW analysis have inferred the distance of Y Oph, but they are scattered between about 400 and 650 pc (Gieren et al. 1993; Kervella et al. 2004a; Mérand et al. 2007; Groenewegen 2008, 2013; Storm et al. 2011). Most of these methods have used the surface brightness-color relation (SBCR). However, Y Oph is close to the galactic plane and exhibits an important color excess of about E(B − V) = 0.650 (Fernie et al. 1995a; Laney & Caldwell 2007; Kovtyukh et al. 2008), which might bias the distance measurements based on photometric relations if the reddening is not well known. Among these BW variants, Mérand et al. (2007) directly observed the angular diameter along the pulsation cycle with infrared (IR) interferometry. It offers the advantage of allowing us to avoid the use of SBCRs which are sensitive to empirical calibrations and interstellar extinction. The derived distance from this method is d = 491 ± 18 pc. On the other hand, the distance obtained from Gaia DR3 parallax (Gaia Collaboration 2016, 2022) is strongly discrepant with d = 742 ± 21 pc (corrected from the zero point, Lindegren et al. 2021). The renormalized weight error (RUWE) of Y Oph indicates a good quality of the parallax (RUWE = 1.04 < 1.4); thus, it can be used for astrometry. Nevertheless, Y Oph slightly saturates the detector with G = 5.6 mag. Indeed, Lindegren et al. (2018) has warned that stars with G < 6 mag generally display an inferior astrometry due to calibration issues. Despite small saturations, we do not expect such large errors of measurements that would explain the difference obtained with BW methods. As an example, T Vul demonstrates both a comparable RUWE and magnitude in G-band with Y Oph, but the distance inferred from its parallax d = 581 ± 20 pc is consistent with HST parallax and BW distance from the literature (Fouqué et al. 2007). Moreover, the color variation of Y Oph is smaller than others Cepheids due to its small light curve amplitude; thus, the chromaticity effects on Gaia detector are likely mitigated.

Because of its low amplitude, Y Oph is sometimes supposed to be a first-overtone pulsator (see e.g., Owens et al. 2022). However, it is questionable to assume that Y Oph pulsates in the first overtone since this mode is usually excited for Cepheids of much shorter pulsation periods. From the observations, the longest pulsation period for a securely identified first-overtone Cepheid is 7.57 days in the case of V440 Per (Baranowski et al. 2009). In the OGLE Milky Way survey, the longest overtone period claimed is 9.437 days (Udalski et al. 2018; Pietrukowicz et al. 2021). Another explanation for the small amplitudes was proposed by Luck et al. (2008). The small amplitudes are not a direct result of the first-overtone mode, but are associated with the proximity to the blue edge of the fundamental mode instability strip. This might be the case for Y Oph which exhibits a mean effective temperature of about 5800 K as measured by spectroscopic observations (Luck 2018; Proxauf et al. 2018; da Silva et al. 2022). This is a high temperature compared to the stars of the same pulsation period, which have a mean temperature of about 5500 K, as in the case of CD Cyg (P = 17.07 days) or SZ Aql (P = 17.14 days), for instance (Trahin et al. 2021).

Several studies have shown spectroscopic binarity evidence of Y Oph, although its effect on photometric and radial velocity measurements remains negligible. Abt & Levy (1978) suggested that Y Oph is a spectroscopic binary system having an orbital period of 2612 days and concluded that its effect on photometry and radial velocity are likely marginal. Szabados (1989) extended the analysis of the mean radial velocity variation and derived an orbital period of 1222.5 days. However, the companion was not detected in IUE spectra (Evans 1992) and NACO lucky imaging yielded a magnitude difference of at least 2.5 mag in the Ks band (Gallenne et al. 2014). More recently, a third companion of much longer orbital period (about 28 000 days) is suspected from light-travel time effect observed in O − C diagram (Csörnyei et al. 2022).

To gain physical insights into this intriguing star, particularly with respect to explaining its low radial velocity and light-curve amplitude, along with the brightness and distance discrepancy, we aim to constrain the Y Oph properties with hydrodynamical pulsation models. Non-linear pulsation modeling offers a valuable approach for exploring the underlying physics of variable stars, as demonstrated for the first time by Christy (1964, 1966a,b,c). Additionally, it provides a tool to accurately determine the mass of Cepheids, a crucial factor when considering the mass discrepancy associated with these stars (Caputo et al. 2005; Bono et al. 2006; Keller 2008). Pulsation models are also useful in determining the distances for Cepheids (Marconi et al. 2013a,b; Ragosta et al. 2019). In the case of Y Oph, a non-linear model was proposed to fit radial velocity measurements and V-band apparent magnitude (Ruoppo et al. 2004). These authors found that a 7 M model is consistent with short distance and low absolute luminosity in V-band (respectively 423 pc and MV = −3.996 mag). However, in this case, Y Oph was modeled with an average effective temperature by about 1000 K lower than recent high-resolution spectroscopic temperature (Teff = 4720 K < 5800 K).

In this paper, we propose, for the first time, to constrain hydrodynamical pulsation models of Y Oph with an extensive set of observations. On the modeling side, the recently released Radial Stellar Pulsation (RSP) tool in Modules for Experiments in Stellar Astrophysics (MESA, Smolec & Moskalik 2008; Paxton et al. 2013, 2015, 2018, 2019; Jermyn et al. 2023) can be used to model non-linear pulsation of Y Oph. MESA-RSP is a one-dimensional (1D) Lagrangian convective code that has already been used for modeling RR Lyrae, BL Her, and classical Cepheids (see, e.g., Paxton et al. 2019; Das et al. 2020, 2021; Kurbah et al. 2023). On the observational side, we benefit from a complete set of observations to constrain the models. To this end, we gathered data from the literature, such as the effective temperature, radial velocity measurements, angular diameter from interferometric observations, and light curves in the visible and the IR.

The paper is organized as follows. We describe our construction of a grid of equilibrium static models and performance of a linear non-adiabatic analysis with MESA-RSP in Sect. 2. Then, we use the solutions that have a positive growth rate for pulsation period of ≈17 days to perform a nonlinear calculation for this star in Sect. 3. We then present our fitting strategy to a full set of observations in Sect. 4. Finally, we present the results in Sect. 5 and we discuss the discrepancy between Gaia and BW distances in Sect. 6. We present our conclusions in Sect. 7.

2. Linear analysis with RSP

Our method consists first of computing a linear non-adiabatic (LNA) stability analysis of a grid of models with MESA-RSP as described, for instance, in Smolec (2016) and Paxton et al. (2019). For this study, we used MESA version r15140 and OPAL opacity tables from Iglesias & Rogers (1996) and Ferguson et al. (2005), using the solar abundance mixture from Asplund et al. (2009). We note that while subsequent MESA version were released after this study, the changes introduced does not affect pulsation calculation with the RSP module (Jermyn et al. 2023). The objective is to determine the metallicity, mass, luminosity, and temperature at which the pulsations are linearly unstable for a pulsation period of approximately 17 days. To this end, we constructed equilibrium static models with a fixed chemical composition from the literature and a grid of mass, M, luminosity, L, and effective temperature, Teff. The results of this computation provide the linear periods of the different radial pulsation modes and their growth rates, γ. The growth rate of a mode γ, is defined by the fractional growth of the kinetic energy per pulsation period.

In the following, we first present our choice for the convective parameters and the chemical composition adopted and we then describe the construction of the grid and the LNA computation.

2.1. Convective parameters adopted

MESA-RSP uses the time-dependent model of turbulent convection described in Kuhfuss (1986) and follows the implementation of Smolec & Moskalik (2008). We used Set A of convective parameters as given in Table 4 in Paxton et al. (2019). Set A represents a simple convection model without radiative cooling, turbulent pressure and flux. We note that in the particular case of a “time-independent” version of Set A, this model would reduced to the standard mixing-length theory (Kuhfuss 1986; Wuchterl & Feuchtinger 1998; Paxton et al. 2019). Furthermore, Paxton et al. (2019) have shown that the set of convective parameters proposed are reasonably reproducing the Fourier parameters of both the radial velocity and I-band light curves of Cepheids (see Fig. 12 in Paxton et al. 2019). On the other hand, the choice of convective parameters also directly impacts the linear period and associated growth rate of the different modes. The introduction of the turbulent pressure, which involves slight inflation of the star, tends to shift the blue edge of the instability strip red-ward (see Fig. 13 in Paxton et al. 2019). Therefore, this will affect the temperature of Y Oph to decrease and modify the growth rates and linear periods associated to fundamental and first-overtone modes. The convective parameters and their respective values have to be properly calibrated and applied with caution (Kovács et al. 2023). However, we do not expect a significant change of the convective parameters used in this study because our modeling is constrained to the observed effective temperature of Y Oph (in average 5800 K), which does not permit the introduction of overly high turbulent pressure, as introduced by Set D (Paxton et al. 2019), for instance. For this same reason, the growth rate associated to different pulsation modes will not change significantly as compared to Set A. While we are confident that Set A of convective parameters is robust to obtain reasonable models of Cepheids, we stress that calibrating these parameters is needed to provide a more precise modeling of Y Oph in the future.

2.2. Chemical composition adopted

A number of studies have derived the metallicity of Y Oph from spectroscopic observations. We summarize the most recent measurements of the iron-to-hydrogen ratio [Fe/H] in Table 1. From this table, we can see that the measurements are in good agreement with solar metallicity. However, the most recent [Fe/H] determination given by da Silva et al. (2022) is not in agreement with earlier estimates.

Table 1.

Spectroscopic metallicities gathered from the literature for Y Oph.

As noted by da Silva et al. (2022), their [Fe/H] determination for several Cepheids is about 0.1 dex below other measurements. In particular, their study estimated the Y Oph metallicity from the same spectra as Proxauf et al. (2018), who found, in turn, a value for [Fe/H] that is shifted by +0.13 dex. According to da Silva et al. (2022), this difference is due to their very careful selection of the lines adopted to estimate the iron abundance. Although more studies are needed to confirm this trend, the recent da Silva et al. (2022) measurement is likely to be the most accurate [Fe/H] value. Since Y Oph’s metallicity is well constrained to the solar abundance, we chose to adopt only a standard solar abundance value for the metallicity of Y Oph that is [Fe/H] = 0.0 dex.

From [Fe/H], we can estimate the mass fraction of hydrogen, X, helium, Y, and metals, Z, related via X + Y + Z = 1. By definition, the metallicity [Fe/H] is given by:

(1)

where X and Z stand for the solar mass fraction of hydrogen and metals, respectively. For these values, we adopted the solar mixture Z/X = 0.134 from Asplund et al. (2009). To derive the helium abundance, Y, in the precedent equation, we assumed the linear relation between Y and Z (Peimbert & Torres-Peimbert 1974):

(2)

where Yp is the primordial helium abundance and ΔYZ is helium-to-metal enrichment ratio. We adopted the primordial helium abundance Yp = 0.2484 ± 0.0005 (Cyburt 2004). Several estimates for ΔYZ can be found in the literature (Tognelli et al. 2021). These values are generally centered on ΔYZ = 2 with large uncertainties. Hence, we chose to explore different ΔYZ values for the adopted solar abundance in the LNA analysis. We chose to extend our grid to ΔYZ = 1.5, 2, and 2.5. Several studies investigated the dependence of helium content at fixed metallicity on the edge of the instability strips derived with non-linear pulsation models (see, e.g., Fiorentino et al. 2002; Marconi et al. 2005).

2.3. Results of the linear non-adiabatic analysis

We chose to perform LNA analysis for six different stellar masses: 3, 4, 5, 6, 7, and 8 M. Our grid is also evenly spaced for the effective temperature by 50 K between 5500 and 6000 K and for the luminosity by 250 L between 3000 and 10 000 L. The temperature grid was chosen in order to cover the average effective temperature measured by Luck (2018) that is 5819 K. On the other hand, the extension of the luminosity grid is chosen to explore the different distance scenarios in the context of the discrepancy between BW methods and Gaia parallax. This ensures that the physical properties of Y Oph are within the limits of the grid. However, we emphasize that it is beyond the scope of this paper to provide a finer resolution of the grid to precisely constrain the physical parameters.

We used a standard grid structure in RSP, which consists of 150 Lagrangian mass shells with variable and constant shell mass, below and above the anchor temperature respectively (see Fig. 2 in Paxton et al. 2019). This zoning was defined to ensure a good spatial resolution of the ionized hydrogen- and helium-driving regions around the anchor temperature of 11 000 K. The effective temperature is attributed to the outer shell, while the inner boundary temperature is set to 2.106 K (see Appendix A.1). This setting is customary in the use of RSP (see, e.g., Paxton et al. 2019; Kovács et al. 2023).

We computed the linear periods of the fundamental mode together with the corresponding linear growth rates. We also derived the linear periods and growth rates for the 11 consecutive radial overtones to investigate the possibility of overtone pulsations for this star. High radial overtones in Cepheids are often trapped in the outer layer of the star. Such trapped modes, dubbed “strange modes” by Buchler, are characterized by very small light amplitudes (Buchler et al. 1997). These modes can be at the origin of non-linear pulsation of Cepheids outside the instability strip on the blue side, which could be of interest in the case of Y Oph. The MESA-RSP script used for the computation is presented in Appendix A.1.

As a result of the computation, we found positive growth rates for the first overtone mode but only for linear pulsation period below about five days. All higher order overtones are linearly damped in our grid. Finally, we can rule out the possibility that Y Oph is a first overtone pulsator or pulsates in even higher overtone modes. The results of the LNA analysis for the fundamental mode are displayed in Fig. 1. From this figure, we can see that a few of these calculated models are missing because the computation failed for numerical reasons, especially at very high temperatures away from the instability strip. This has no impact on the remainder of the study.

thumbnail Fig. 1.

Results of the linear non-adiabatic (LNA) analysis with MESA-RSP. Positive and negative growth rates, γ for the fundamental mode are indicated in red and grey points, respectively. The black crosses are the selected models for non-linear analysis as summarized in Table 2. The vertical blue strip corresponds to the expected value of the mean effective temperature for Y Oph, i.e., Teff = 5800 ± 100 K (Luck 2018; Proxauf et al. 2018; da Silva et al. 2022). The fundamental blue edge from Anderson et al. (2016) is shown for comparison (see their Fig. 2: Z = 0.14, ωini = 0.5, mix between 2nd and 3rd crossing) with ΔL ± 500 L.

To find models with positive growth rates for the fundamental mode with period of P ≃ 17.12 days, we first interpolated growth rates along the luminosity axis. Then, we interpolated the pulsation period along L and Teff to determine an iso-period line at which the models pulsate with the period close to P = 17.12 days, as observed in the star (see orange lines in Fig. 1). The intersection of the iso-period line and the positive growth rates region provides the combination of mass, metallicity, luminosity, and effective temperature we can consider for a non-linear modeling of Y Oph pulsations. We note that we adopted the non-linear pulsation period to constrain the linear period from the grid. In principle, these two values are not equal for a same model, the non-linear value being slightly longer than the linear period. For the sake of simplicity, we have chosen to neglect this difference which is at the percent level. For each set of mass, we selected models on iso-periods evenly spaced by 25 K (see black crosses in Fig. 1). For each mass, these combinations of effective temperature and luminosity will be then used for a non-linear analysis in the next section (see Table 2). From our computations, we do not observe any significant difference of luminosity and temperature from the different adopted values of ΔYZ. This is in agreement with non-linear computation results from Marconi et al. (2005), who found no clear trend on the fundamental blue edge with helium content change. Hence, we fixed ΔYZ = 2 and we used the grid computed in Table 2 in the following section.

Table 2.

Models selected for non-linear computations, based on results of the LNA analysis for a linear period of P ≃ 17 days (see Fig. 1).

3. Non-linear analysis with RSP

3.1. Non-linear computations

Full-amplitude stable pulsations are reached when kinetic energy per pulsation period becomes constant. In other words, a fractional growth of the kinetic energy per pulsation period, Γ, is equal to 0. Given a running window, our models verify the following conditions:

  • Period stability: the ratio of the standard deviation of the pulsation period to the average pulsation period is less than 10−5,

    (3)

  • Amplitude stability: standard deviation of the radius amplitude to the average radius is less than 10−4,

    (4)

  • Kinetic energy stability: the change of the absolute kinetic energy per pulsation period is below 10−5,

    (5)

Once these criteria are verified altogether, we let the computations run for 200 additional pulsation cycles. For each model we retrieved the two last pulsation cycles computed. Given our time resolution, this corresponds to about 2400 data points. The results along the pulsation cycle are then phased to ϕ = 0 for maximum light of the V-band light curve.

3.2. Atmosphere models along the pulsation cycle

From the non-linear computation, MESA/RSP provides bolometric corrections to obtain the photometry in specific bands of the Johnson-Cousins system (Lejeune et al. 1998). However, since we retrieved the light curves observations in different filters (Sect. 4.1) we preferred to apply the specific synthetic filters to a grid of atmosphere models derived for every phase of non-linear model. The radius and effective temperature are defined by RSP following the Stefan–Boltzman law, (Smolec & Moskalik 2008; Paxton et al. 2019), making them convenient for computing static atmosphere models. To this end, for each effective temperature, Teff(ϕ), radius, RRSP(ϕ), and log g(ϕ) computed by MESA-RSP along the pulsation phase ϕ, we interpolated over ATLAS9 model grid to obtain the spectral energy distribution of these models. We used ATLAS9 models1 (Castelli & Kurucz 2003) with solar metallicity and a standard turbulent velocity of 2 km s−1. We note that the quasi-static assumption of the atmosphere of Y Oph is particularly well adapted as a result of both small amplitude and long pulsation period. Then, we applied the synthetic filters corresponding to the observations in VJHKsLM-bands presented in the next section. These synthetic filters (Johnson, 2MASS and Spitzer IRAC) are publicly available from the Spanish Virtual Observatory2 (SVO).

4. Fitting strategy

4.1. Set of observations

To assess which models are the best to reproduce Y Oph characteristics, we compare the results with a full set of observations. We gathered the following set of observations along the pulsation cycle:

  • Uniform disk angular diameters along the pulsation cycle are provided by near-infrared (NIR) interferometric observations in the K-band from the CHARA/FLUOR instrument (Mérand et al. 2007; Gallenne 2011). The angular diameter measurements are displayed in Fig. 2a.

  • Radial velocities measurements were retrieved from Petterson et al. (2005), Borgniet et al. (2019), Eaton (2020) and phased together and corrected for zero-point offsets. The RV measurements are displayed in Fig. 2b.

  • The effective temperature measurements were obtained from high-resolution spectroscopy by Luck (2018) and da Silva et al. (2022). The measurements obtained by Luck (2018) cover most of the pulsation cycle, whereas those obtained by da Silva et al. (2022) are close to the minimum temperature. The combination of these two independent measurements is in excellent agreement, as we can see from the temperature plot in Fig. 2c.

  • The light curve in the V-band was obtained from the Johnson filter (Berdnikov 2008). These observations are a compilation of many observations between MJD = 46 000 and 53 000. To mitigate the effet of phase mismatch, we selected V-band photometry of MJD before 48 000 to match the observation epoch of the NIR observations. We retrieved the light curves in J, H, and K from Laney & Stobie (1992). We transformed these photometries from the South African Astronomical Observatory (SAAO, Carter 1990) system into the Two Micron All Sky Survey (2MASS, Skrutskie et al. 2006) system using transformations from Koen et al. (2007). This transformation allows us to use the interstellar extinction model calibrated with this same system (see Sect. 4.2). We complemented these data with mid-IR light curves obtained from the Infrared Array Camera (IRAC) on board the Spitzer telescope (Monson et al. 2012) at 3.5 and 4.5 μm (i.e., the L and M bands). These light-curves are displayed in Figs. 2d–i.

thumbnail Fig. 2.

Best result for the non-linear analysis with RSP/MESA for 8 M model. Uniform disk angular diameter, radial velocity curve, and effective temperature are displayed in (a), (b), and (c), respectively. The photometric panels indicate the apparent magnitudes in (d) V-band, (e) J-band, (f) H-band, (g) KS-band, (h) L-band, and (i) M-band. In the angular diameter and photometric panels, the thick black line and dashed grey lines are the RSP models, with and without the CSE models, respectively.

The rate of period change of Y Oph is known to be about +8 s yr−1 (Fernie et al. 1995b; Csörnyei et al. 2022). However, an in-depth analysis of O–C diagram reveals a wave-like signal superimposed to the evolutionary parabolic trend (Csörnyei et al. 2022), as presented in Fig. 3. Unfortunately, our sequence of observations used in this paper falls exactly in the wave-like signal observed in the O − C diagram (see blue strip in Fig. 3). To correct for these effects, we took into account the O − C correction modeled by a parabola and we also applied a second order correction corresponding to a linear fit of the O − C residuals in our observation range (see Fig. 3). In order to phase the observations to maximum light in V-band, we used reference epoch, T0 = 39 853.30, and pulsation period, P = 17.12413 days, from Samus’ et al. (2017). We authorize a slight phase shift (δϕ = 0.02) to perfectly align at the phase of maximum light.

thumbnail Fig. 3.

O − C diagram of Y Oph. O − C calculations were performed by Csörnyei et al. (2022). The dashed line is a parabola fit over the entire range of data. The continuous line on the bottom plot is the linear fit of the O − C residuals over the observation period used in this paper.

4.2. Modeling interstellar extinction

Since we used photometric bands in multiple filters from the visible to the IR, we must define a consistent model to correct for interstellar extinction in the line of sight of Y Oph. We chose a standard total-to-selective extinction ratio, RV = AV/E(B − V) = 3.1, corresponding to an average extinction law of diffuse interstellar medium along the line of sight (Savage & Mathis 1979; Wang et al. 2017). In the IR regime, the extinction law is approximated by a power law of Aλ ∝ λα. However, as highlighted by Wang et al. (2017), Nogueras-Lara et al. (2019) there is a diversity of extinction curves in the diffuse interstellar medium and towards the Galactic Center. The determination of the extinction coefficient α increased significantly in the past decades from about 1.5 (Rieke & Lebofsky 1985; Cardelli et al. 1989) up to a steeper slope with α > 2.0 (see Nogueras-Lara et al. 2019, and references therein). Among the variety of Milky Way extinction curves available in the near- and mid-IR (e.g., Cardelli et al. 1989; Indebetouw et al. 2005; Nishiyama et al. 2009; Wang et al. 2017; Chen et al. 2018), we chose to adopt the interstellar extinction law from Nishiyama et al. (2008, 2009), calibrated from 2MASS and Spitzer/IRAC observations (consistently with our photometric bands, described in Sect. 4.1).

4.3. Modeling the circumstellar envelope

4.3.1. Impact on the photometry

In the case of classical Cepheids, it is often difficult to fit all the photometry in different bands at once from a simple atmosphere model because of the degeneracy between distance, interstellar extinction, and circumstellar envelope (CSE) emission and absorption. Unfortunately, the impact of CSE on Cepheid photometry is often omitted and might be a source of systematic error on individual distance and color excess measurements. The CSEs were resolved from visible and IR interferometry around several Galactic Cepheids (Kervella et al. 2006, 2009; Mérand et al. 2006, 2007; Gallenne et al. 2011, 2013; Nardetto et al. 2016; Hocdé et al. 2021). In the case of Y Oph, Mérand et al. (2007) showed that a CSE emission of about −0.05 mag in the K-band can explain the bias between CHARA/FLUOR and VLTI/VINCI interferometric observations (Kervella et al. 2004b). Furthermore, Gallenne et al. (2012) also detected IR emission from photometry analysis and modeled a significantly hotter CSE than other Cepheids. The IR emission could be caused by a shell of ionized gas around the Cepheids (Hocdé et al. 2020a,b) or a dust envelope (Gallenne et al. 2013; Groenewegen 2020a). However, an excess in the NIR is likely only caused by breaking radiation from a hot ionized gas envelope (Hocdé et al. 2020a), as the dust is too cold to produce significant emission. In the case of ionized hydrogen opacity, absorption can be caused by bound-free processes in the optical, while free-free emission dominates in the IR. The IR excess exhibits an asymptotic behaviour with longer wavelength because the ionized gas becomes optically thick (Hocdé et al. 2020a). Previous studies modeled the IR excess with a parametric power law, assuming that there is no excess, nor deficit in the visible range (Mérand et al. 2015; Trahin et al. 2021; Gallenne et al. 2021). In order to analytically model this process in a convenient way, we introduce the following logistic function:

(6)

where α and β represent the intensity and the slope of the logistic function, respectively, and λ0 is the pivot wavelength for which Δmagλ0 = 0. Although this parametric model certainly provides more physical justification compared to a simple power law, we note that it cannot reproduce complex features from bound-free absorption, a task that can only be accomplished by radiative transfer models.

4.3.2. Impact on angular diameter measurement

The CSE model has also an impact on the measured uniform disk (UD) angular diameter θUD via interferometry. Mérand et al. (2007) have shown that θUD measurements are slightly biased toward larger diameter in the K-band because the CSE emission is partially resolved. More precisely, the bias is k = θUD/θ ≈ 1.023 in the case of Y Oph (Mérand et al. 2007), where θ = θLD is the limb-darkened angular diameter. In principle, it is necessary to assume a CSE geometry and opacity to derive the bias on θUD on the CHARA/FLUOR band. For the sake of simplicity, in the following, we assume a bias of k = 1.023 to take the CSE into account (Mérand et al. 2007).

4.4. Fitting process

Our fitting approach consists of adjusting simultaneously the distance, color excess, and CSE parametric model to fit the angular diameter and the photometries.

Indeed, the photosphere radius RRSP derived by MESA-RSP allows us to unambiguously determine the distance via the angular diameter observations, following:

(7)

Simultaneously to the angular diameter, we fit the entire set of observed light curves in the VJHKSLM bands. To this end, we converted the absolute magnitudes Mλ(ϕ) in each photometric band along the pulsation cycle into apparent magnitudes, following:

(8)

where Aλ is the interstellar absorption in each band and Δmagλ(α, β, λ0) is the absorption or extinction produced by the parametric CSE model as a function of the effective wavelength λ of each synthetic filter (see Eq. (6)).

To summarize, we simultaneously fit Eqs. (7) and (8), adjusting the free parameters dpc, E(B − V) and the CSE model parameters, α, β, and λ0. We performed reduced χ2 statistics from Python Kapteyn package (Terlouw & Vogelaar 2014), which makes use of the Marquardt–Levenberg algorithm (Levenberg 1944; Marquardt 1963) to solve the least-squares problem. For every quantity fitted in our study, we derived statistical errors using the bootstrap method. The fitting of each light curves is displayed in Fig. 2 and in Figs. B.1B.5. The resulting CSE model is displayed in Fig. 4 and the quantitative results are provided in Table 3.

thumbnail Fig. 4.

Fit of the parametric CSE model derived from Eq. (6) in the case of the best fit at 8 M. Errors bars represent the standard deviation of photometric measurements compared to the model. The grey region is the error on the magnitude obtained using the covariance matrix of the fitting result.

Despite small statistical uncertainties of the order of a few parsecs, as presented in Table 3, our distance calculation might be biased by larger systematic errors due to the assumptions of our modeling. For example, the derived radius might be sensitive to the various choices of hydrodynamical modeling parameters. It is difficult to estimate the systematic uncertainty of the derived distance without in-depth pulsation modeling. Hence, for the rest of the study, we arbitrarily assumed an uncertainty of ±15 pc.

In this method, we note that we did not take into account the RV curves because it requires fitting the p-factor to transform RSP pulsation velocity, (Vp), into RV measurements, (VRV), following Vp = pVRV. This is possible in principle, as was done by Marconi et al. (2013b), for example. However, this method is relevant only if the light amplitude modeled from MESA/RSP is adjusted in order to fit the observed light amplitude. Since we did not perform a fit, but simply compared the observations to a grid of models, we cannot include the RV curves in the calculation. For each model, however, we simply assume p = 1.27 and present the result for consistency only.

Finally, in order to assess which models offer the best fit to the observations, we computed the total to take into account the following observations along the pulsation cycle: light curves, angular diameter, and effective temperature. To this end, the total is derived as the average χ2 for all observables. Hence, each observable contributes equally to the final likelihood estimation. The results are presented in Table 3. For each mass, we displayed as a function of the effective temperature in Fig. 5.

thumbnail Fig. 5.

Mean versus the mean effective temperature for models of different masses.

Table 3.

Best fits of the non-linear models.

5. Results

5.1. Proximity with the blue edge of the IS

The first striking feature of the non-linear computations is that there is a remarkable agreement between the measurements along the pulsation cycle and almost all models at the highest effective temperature. This is summarized in Fig. 5, where we plot variations of the total χ2 versus the effective temperature. We also emphasize the overall success of these pulsation models in simultaneously reproduce many different types and complex observations along the pulsation cycle remarkably well. Indeed, we observe that the best RSP models all closely follow the variations of angular diameter measured by interferometry, radial velocity, and effective temperature obtained by high-resolution spectroscopy and light-curves in VJHKsLM bands. Hence, our first conclusion is that the quasi-symmetric and low amplitude of the different curves are attributed to the extreme proximity of Y Oph with the blue edge of the instability strip (IS), independently of the assumed stellar mass adopted.

5.2. Distance discrepancy

From the results of our non-linear computations, it is not possible to determine the best mass for Y Oph based only on the shape of the different curves. Indeed, even if the best value is obtained for the 7 M models at Teff = 5750 K, all the other best models are essentially equivalent regarding the various modeling assumptions and observational uncertainties.

Nevertheless, the derived distance for each mass can be used to discern the best model. In Table 4 and Fig. 6a, we provide a comparison of the different distances from the literature, together with the distance we derived from RSP models for each mass. As we can see, the different variants of the BW method yield distances in agreement with the lower stellar mass models for Y Oph. In particular, the interferometric BW distance (hereafter IBW) from Mérand et al. (2007) is in excellent agreement with the RSP distance obtained for a stellar mass of about 3 M. On the other hand, the Gaia distance yields a stellar mass of at least 7 M according to our modeling. This result is in agreement with masses inferred from pulsation models constrained by Gaia parallax of Cepheids, with similar pulsation period (Marconi et al. 2020; De Somma et al. 2020); although a slightly higher mass was derived in the case of Y Oph (10.4 ± 1.4 M). Last, the luminosity as predicted by higher stellar mass models are in good agreement with PL relations in the KS band (see Fig. 6b); whereas this is discrepant for lower stellar mass, in particular, for 3 M.

thumbnail Fig. 6.

Comparison of distance and absolute luminosity in the Ks band between the literature and our best results from non-linear models. (a) Comparison of distances obtained using variant of SBCR Baade-Wesselink methods based on (V − R) (Gieren 1988; Hindsley & Bell 1989; Gieren et al. 1993; V − K) (Storm et al. 2004, 2011; Barnes et al. 2005; Groenewegen 2008, 2013); interferometric Baade-Wesselink method (Mérand et al. 2007; Kervella et al. 2004a); distance obtained from the parallax measured by Gaia DR2 and DR3 (Gaia Collaboration 2022); and distance inferred from RSP models (see Table 3) as vertical strips for each stellar mass assuming uncertainty of ±15 pc. The average of both BW distances based on (V − K) color and their p-factor are indicated. (b) Comparison of KS-band absolute magnitude derived from different PL relations (Benedict et al. 2007; Fouqué et al. 2007; Gieren et al. 2018; Groenewegen 2018; Breuval et al. 2020, 2021; Trahin et al. 2021) established in the 2MASS filter (see Breuval et al. 2020), and the magnitude as we derived in the 2MASS filter from MESA-RSP non-linear models.

Table 4.

Summary of Baade–Wesselink distances for Y Oph from the literature.

5.3. Color excess and CSE model

For every fit we computed, we observed a range of E(B − V) from about 0.620 and 0.670. These values are in excellent agreement with color excess measurements from the literature: 0.645 ± 0.032 mag (Fernie et al. 1995a)3; 0.660 ± 0.02 mag (Laney & Caldwell 2007) and 0.683 ± 0.01 mag (Kovtyukh et al. 2008). To check the consistency of the color excess with the derived distance, we compared our result with 3D extinction map in the Galaxy from Lallement et al. (2014), Capitanio et al. (2017) in Fig. 7. In this figure, we observe that the Gaia DR3 distance is in agreement with E(B − V) = 0.600 ± 0.15 mag. Our best 8 M model is also consistent with both the Gaia DR3 distance and color excess measurements (see dashed lines in Fig. 7). On the contrary, shorter distances derived from BW measurements are only in marginal agreement with 3D extinction map. In particular, distances below 500 pc (Gieren 1988; Gieren et al. 1993; Mérand et al. 2007) are not consistent with a high color excess of 0.600 mag or more. This result suggests that the color excess derived in our calculation is consistent only for distance higher than about 600 pc which excludes almost all distances derived from BW variants (see Fig. 6a). The choice of a near- and mid-infrared extinction law in Sect. 4.2 does not change our main result, which is the derived distance for each model. Indeed, the distance depends almost solely on the radius and angular diameter measurement. However, it has an impact on the derived shape of the CSE model adopted.

thumbnail Fig. 7.

Color excess E(B − V) in the direction of Y Oph versus the distance, from 3D extinction map (Lallement et al. 2014; Capitanio et al. 2017). The horizontal red strip indicates mean E(B − V) and standard deviation of the mean as compiled by Fernie et al. (1995a). Green and blue strips indicate the mean BW(V − K) distance and dGaia distance Gaia DR3 (Gaia Collaboration 2022), respectively.

As expected, the CSE absorption in the visible range is an additional source of extinction. We derived a CSE absorption in the visible of about 0.10 mag (see Fig. 4). This level of absorption is in agreement with bound-free absorptions derived by Hocdé et al. (2020a) from radiative transfer models. If we recalculate the fit, but omitting the modeling of the CSE in Eq. (8), we still obtain the same distance, but the extinction compensates for the absence of CSE in the visible. For example, in the case of the 8 M model, we find E(B − V) = 0.666 ± 0.003 mag, which is 0.033 mag larger. We obtained a pivot wavelength λ0 between 1.5 and 2 μm which is also consistent with radiative transfer of ionized gas (Hocdé et al. 2020a). In the IR, we obtain a small excess in the K-band close to the CSE emission derived from FLUOR/CHARA observations (Mérand et al. 2007). At longer wavelengths, the IR excess stabilizes at about −0.15 mag, which is in agreement with the results of Gallenne et al. (2012), who found an excess of 15% compared to the stellar photosphere at 8.6 μm for Y Oph. This confirms the importance of modeling the CSE in the framework of adjusting atmosphere and pulsation models to visible and IR observations (see Figs. 2d–i).

6. Discussion

6.1. Comparison with evolutionary masses

In order to compare our results with evolutionary models, we computed the evolutionary tracks at solar metallicity for the same helium abundance and sample of stellar masses with MESA. The use of MESA for this comparison is most appropriate, since both MESA and MESA-RSP are based on the same implementation of microphysics and use the same numerical algorithms. We considered non-canonical models with an overshooting parameter fov = 0.02, without rotation and no mass loss. We chose this model to simply illustrate the consistency of evolutionary models with our pulsation modeling. More detailed investigations are necessary but beyond the scope of this paper that is dedicated to the pulsation. Other details of this model will be given in paper in preparation (Ziółkowska et al., in prep.). Comparing MESA evolutionary tracks with MESA-RSP pulsation models (see Figs. 8a and b), we observe that only the blue loops of the higher mass models at 8 M can reach the luminosity of the RSP models. In other words, a simple non-canonical evolutionary model with moderate convective overshooting is able to explain simultaneously the pulsational mass derived for Y Oph and the Gaia distance. The rate of period change of Y Oph +8.14 ± 0.25 s yr−1 (Csörnyei et al. 2022) is consistent with typical values of long-period Cepheids in the 3rd crossing of the instability strip (Turner et al. 2006). The rate of period change for stars of similar pulsation period in the first crossing is at least one order of magnitude larger. We can thus rule out the possibility that the star is on its first crossing.

thumbnail Fig. 8.

HR diagram comparing parameters of MESA-RSP models of Y Oph given in Table 2 (crosses) with evolutionary tracks computed with MESA for different stellar masses (MESA version r-21.12.1 used by Ziółkowska et al., in prep.). These models assume an overshooting parameter fov = 0.02. The vertical blue strip corresponds to the measured value of the mean effective temperature for Y Oph, i.e. Teff = 5800 ± 100 K.

On the contrary, Fig. 8 also shows that the pulsation models with masses of 3 and 5 M, namely, masses derived for the BW distances, are overluminous, as compared to the corresponding evolution tracks. In fact, the blue loops of 3–4 M evolutionary tracks do not even penetrate the Teff range of Y Oph, as also shown by other evolutionary computations (Anderson 2014; De Somma et al. 2021).

We note that the presence of CSE could indicate significant prior mass loss, although we did not take this phenomenon into account in our evolutionary models. In this case, Y Oph could be produced by a higher prior mass Cepheid. This is not surprising, as mass loss is one of the ingredient to solve mass discrepancy of Cepheids. The range of mass-loss rates generally attributed to Cepheids are between 10−10 and 10−6M yr−1 (see e.g., Deasy 1988; Neilson & Lester 2008; Gallenne et al. 2013). In the case of Y Oph, Gallenne et al. (2012) derived a minimum mass-loss of 2 × 10−10M yr−1 assuming that the mass loss is driven by radiation pressure on the dust envelope. Although there is no consensus on the mass-loss mechanism, it is expected that pulsation-driven mass loss might be the most efficient (Neilson & Lester 2008; Neilson et al. 2012). The effect of mass loss is likely not significant on our pulsation modeling, which represents a too short time interval for the Cepheid, but the structure of the blue loop as computed by evolutionary models might be affected. However, evolutionary codes implemented simple empirical models such as Reimers (1977) and de Jager et al. (1988) that are used by default to study Cepheid mass-loss. Unfortunately, these models are not only inefficient, but also not relevant for Cepheids for which pulsation likely plays a crucial role in driving the mass-loss. Therefore, the effect of mass-loss on Cepheid evolution remains to be understood to consistently compare the pulsation and evolution models. More detailed evolutionary calculations will be necessary to better understand the evolutionary state of Y Oph.

6.2. BW distances versus Gaia distance

If we assume that the Gaia parallax is accurate for Y Oph, the question is now to understand why all BW distances are discrepant (see Fig. 6a). For every BW distance presented in this paper, we note the heterogeneity of these analyses in terms of fitting procedure, source of radial velocity, and the angular diameter measured by interferometry or deduced by SBCRs, as well as the photometry used and color extinction, as presented in Table 4. All these elements impact the distance measurements differently (see e.g., Nardetto et al. 2023) and might explain why BW distances are scattered. Overall, the mean BW distance derived from these methods is 581 ± 24 pc for an average p-factor p = 1.29 ± 0.08 (see Fig. 6a). This average distance is clearly in disagreement with our computed models at 7 and 8 M and Gaia distances (DR2 and DR3) and would require a p-factor above 1.5 (formally (742/581) × 1.29 = 1.65). Before any conclusions on the causes of discrepancies between BW and Gaia distances are drawn, we must first scrutinize the SBCR used for the Y Oph distance determination.

6.3. Impact of the CSE on the SBCR

The SBCR is critical in BW methods to determine the distance of Y Oph. We investigate the SBCR of Y Oph using a theoretical SBCR based on the 8 M RSP non-linear model derived previously:

(9)

where Teff and the bolometric correction in the visible BCV are directly given by the output of the RSP model, without any hypothesis needed on the distance, the extinction, nor the CSE model. We then plotted this value against (V − K)0, where V and K are the absolute magnitude derived from atmosphere models interpolated from RSP stellar parameters (see Sect. 3.2). As we can see from Fig. 9, we obtain an excellent agreement with empirical SBCR relation of Cepheids derived by Kervella et al. (2004b). A linear fit based on the SBCR derived with RSP gives .

thumbnail Fig. 9.

Comparison of SBCR derived for Y Oph with RSP along the pulsation cycle and empirical SBCR from Kervella et al. (2004b).

At first glance, there is no significant difference between theoretical SBCR for Y Oph as derived from RSP and the fiducial SBCR of Cepheids. However, Groenewegen (2007, 2013) has shown that the “empirical” SBCR derived for Y Oph is discrepant with the relation calibrated by Kervella et al. (2004b). In particular, this star appears too red at a given surface brightness or, conversely, the surface brightness is too high at a given color. Until now, the origin of this discrepancy was attributed to uncertainty of the interstellar extinction. Groenewegen (2013) argued that a value of E(B − V)≈1 would be enough to bring Y Oph onto the relation. We show hereafter that the CSE model derived in this paper for Y Oph is able to give an interesting alternative explanation without assuming exceptionally large E(B − V) (see also Fig. 7 for comparison). To check for the CSE impact, we reproduced the SBCR “observed” for Y Oph by omitting the existence of the CSE:

(10)

where θLD is the limb-darkened angular diameter obtained from θUD observed by interferometry (corrected for LD effect) and V0 is the dereddened apparent magnitude in the visual band. Dereddened apparent magnitudes in V and K bands, interpolated to the phase of angular diameter measurements, are also used to compute (V − K)0. We stress that we deliberately did not correct for the influence of the CSE on the apparent V and K magnitudes and the angular diameter measurements, so that we have the ability to make comparisons with prior studies (see, e.g., Groenewegen 2013).

As a result, the empirical SBCR of Y Oph appears to be significantly higher than the fiducial SBCR, although within the uncertainty (see red points in Fig. 9). This result is in agreement with the discrepancy previously observed by Groenewegen (2007, 2013). Finally, if now we add the CSE model presented in Fig. 4 (as derived in Sect. 4.4) on the theoretical SBCR, we obtain a result that is in agreement with observations (see red line in Fig. 9).

Now the question is to quantify the impact of the CSE on the derived distance when using SBCR relation. We define the CSE effect on the visible and the IR magnitudes as ΔV and ΔK, respectively. The observed magnitudes of the star that are dereddened, but not corrected for the presence of CSE, can be expressed as and , where V0 and K0 are the dereddened magnitudes of the star itself. We can then substitute these magnitudes into the following equations that are used to derive the angular diameter:

(11)

(12)

The combination of these equation allows us to derive the CSE impact on the distance:

(13)

In the case of an absence of CSE, ΔV = ΔK = 0, there is obviously no bias on the derived distance. If there is a CSE absorption and emission in the visible and IR, respectively, then ΔV > 0 and ΔK< 0. Thus, both terms of the above equation are contributing to lower the derived distance. In the case of our CSE model, we have ΔV = 0.10 mag and ΔK = −0.05 mag, which translates into a distance 5% lower (note: we used a = −0.133 consistently to Kervella et al. 2004b). We find that the IR emission from the CSE has the most significant impact on the SBCR, in agreement with the findings of Nardetto et al. (2023). As a conclusion, we have shown that the CSE of Y Oph might be a significant bias on the SBCR. Therefore, studies which made use of any calibrated SBCRs to derive the distance of Y Oph very likely underestimated the surface brightness of Y Oph. The unbiased average BW distance, namely, corrected from CSE effect, is thus closer to 610 ± 24 pc. Although the tension with Gaia distance is relaxed, the discrepancy is still at the 4σ level. We discuss the possibility of a high p-factor in Sects. 6.5 and 6.6.

6.4. Comment on SBCR based on V − R

Using Eq. (13) we can estimate the systematic error on the distance in SBCR using V − R color. We find that the derived BW distance must be overestimated by a few percent because of the presence of the CSE. Therefore, it is difficult to explain why Gieren (1988) and Gieren et al. (1993) found much smaller distance than subsequent BW methods. Several independent reasons may explain this difference. First, Gieren et al. (1993) noticed that Y Oph is biased compared to their calibrated SBCR relation, which simply invalidates their distance measurement for Y Oph. Second, the latter studies use RV data from Coulson & Caldwell (1985), which is not accurate enough and suffers from a smaller amplitude by 20%, as noted by Hindsley & Bell (1989). Last, they calibrate (V − R) on the basis of atmosphere models, which has been later proven to be biased by Fouque & Gieren (1997). Curiously, despite the large uncertainties, Hindsley & Bell (1989) found a mean distance that is in excellent agreement with Gaia observations. We think this result is primarily accidental, as most of their distance determinations of Cepheids are strongly biased towards larger distance.

6.5. Discussion of the high p-factor

Although BW distance measurements presented in Fig. 6a were probably underestimated, the BW methods (corrected by CSE) can only be matched with Gaia DR3 distances if (in turn) the p-factor is about p = 1.5. Qualitatively, a high p-factor (but below 1.5) is associated to a lower velocity gradient in the atmosphere or a weaker limb-darkening effect (Nardetto et al. 2006). This is not usually the case for long-period Cepheids which have larger velocity gradient and large limb-darkening effect (Nardetto et al. 2006); however, this could be a good guess for Y Oph regarding its small light and radial velocity amplitude. Moreover, large values slightly above 1.5 were also occasionally reported in the literature for different Cepheids (Storm et al. 2011; Trahin et al. 2021), but they can be in general attributed to observational uncertainties. It is physically conjectured that a reverse atmospheric velocity gradient or a limb-brightening effect can bring the p-factor above 1.5 (Albrow & Cottrell 1994; Sabbey et al. 1995; Nardetto et al. 2006; Storm et al. 2011; Ngeow et al. 2012; Neilson et al. 2012). A limb-brightening effect might be caused, for instance, by the presence of a chromosphere (Neilson et al. 2012). In addition, Sabbey et al. (1995) found p = 1.6 from radiative hydrodynamical models during the expansion of the photosphere, which suggests a limb-brightening of the spectral lines. However, the absence of Ca II K emission for Y Oph should (in principle) exclude any strong chromospheric activity (Kraft 1957).

Paradoxically, Wallerstein et al. (1992) concluded that Y Oph would be more reliable to use for BW analysis because of the smooth and synchronized variation of Hα and metallic lines observed for this star. In contrast, Albrow & Cottrell (1994) argued that in the case of low pulsation velocity in combination of radial macroturbulence, the central intensity profiles are broadened and the major contribution of the radial velocity comes actually from region closer to the limb of the stellar disk. As a result, the lines that have an opposite asymmetry are able to produce a larger p-factor. In conclusion, we suggest that Y Oph has peculiar photospheric characteristics that makes it extremely interesting for studying the physics behind the p-factor.

6.6. Interferometric BW distance versus Gaia distance

We might argue that the SBCR is not considered in the interferometric Baade–Wesselink distance determination from (Mérand et al. 2007, hereafter, M07) and, thus, an even larger discrepancy remains with the Gaia distance, since the IBW distance is particularly small compared to other methods (see Fig. 6a). However, a simple test is enough to show that angular diameter distance measurements from M07 can be also explained by a high distance in combination of p-factor limited by the geometry, as we can see in Fig. 10. To this end, we performed a RV curve fitting using observations presented in Sect. 4.1. We derived the angular diameter based on two cases, the first one taking into account p = 1.27 (similarly to M07) and the second one assuming the Gaia distance together with a p-factor limited by the geometry (p = 1.5). We used the following equation which transforms radial velocity into angular diameter variation:

(14)

thumbnail Fig. 10.

Comparison of uniform disk angular diameter measurements in K-band from Mérand et al. (2007) with a combination of the Gaia DR3 distance and a p-factor of 1.5. The orange bar is a discrepant measurement compared to Gaia distance model (see Sect. 6.6).

where θUD is the interferometric uniform disk diameter and k is defined as previously k = θUD/θ = 1.023, consistently with M07. In the first case, we chose p = 1.27, as assumed in M07 and we fit the distance and the mean angular diameter, θUD(0). We find a distance of d = 480 pc with a reduced chi-square = 0.38, in close agreement with the results obtained by M07 and Gallenne (2011). In the second case, we fixed the p-factor to p = 1.5 and the distance to Gaia DR3 d = 742 pc, and we adjusted the mean angular diameter only. We obtained = 0.76. Although the distance from M07 has the smallest χ2, the goodness of the fit obtained from the Gaia distance together with a high p-factor has also an excellent likelihood. Moreover, if we remove the most discrepant measurement (see orange error bar in Fig. 10), we essentially obtain the same likelihood for both models with = 0.41 versus 0.48 for M07 and the Gaia distance, respectively.

Interestingly, Kervella et al. (2004a) derived a distance of d = 648 ± 50 pc which is the closest BW distance to the Gaia observation. This later result was established using an hybrid version of the BW method, where a value of the linear radius was assumed to be about 100 R using a canonical Period–Radius relation. Therefore, it is not surprising that this determination is closest to the Gaia distance. However, we think that there are two issues preventing this method from perfectly matching the Gaia distance. First of all, the p-factor used in this method is probably too low (p = 1.36), as we argue in the previous section. Secondly, the VINCI/VLTI uniform disk angular diameter is likely to be too large because it resolves the CSE emission in the K-band, as shown by Mérand et al. (2007). This effect tends also to derive a lower distance. Therefore, this test supports our result that all BW distances of Y Oph can be explained by the combination of CSE impact and high p-factor.

6.7. Systematic errors of temperature measurements

The effective temperature is a critical parameter of our modeling since the high average temperature of Y Oph of about 5819 K (Luck 2018) places the star close to the blue edge of the instability strip. Therefore, it is important to discuss the possibility of systematic errors of the effective temperature measurements. The effective temperature used in this study was determined using the line depth ratio (LDR) for several pair of absorption lines from Luck (2018) and da Silva et al. (2022). According to Kovtyukh et al. (2008), although Teff are provided with a small internal error (typically 100 K or less), a systematic error could exist. Furthermore, da Silva et al. (2022) re-analyzed HARPS spectra used by Proxauf et al. (2018). As explained in Sect. 2.2, da Silva et al. (2022) focused on possible systematics affecting the estimation of atmospheric parameters and they developed a careful analysis of the adopted lines. Although they derived a systematically lower [Fe/H], the mean Teff derived for Y Oph is in agreement with the determination of Proxauf et al. (2018) at a minimum temperature of the pulsation cycle (5609 ± 33 K versus 5612 ± 33 K). Moreover, as we noted previously, temperature measurements of Luck (2018) and da Silva et al. (2022), which are based on different sets of spectra, are also in agreement along the pulsation cycle. Therefore, we are confident that Teff determination of Y Oph is accurate.

The presence of systematic errors in temperature determinations remains however a topic of discussion (Mancino et al. 2021). To investigate possible systematic effects, Groenewegen (2020a,b) compared the average Teff determination of Cepheids from Luck (2018) (for 52 stars with more than 5 spectra) to photometric temperature using SED fitting. Interestingly, Groenewegen (2020b) found an agreement within the uncertainties for all stars except Y Oph and S Vul. The photometric temperature of Y Oph is found to be 570 K lower than spectroscopic temperature. However, photometric temperatures based on SED fitting is mostly sensitive to the optical data (Wien’s law). Hence, this method can be strongly biased because of CSE visual absorption and the color excess of this star. If the CSE absorption is ignored during the fit, then its contribution is hidden in the fitted E(B − V) term. On the contrary, if E(B − V) is fixed to literature values, which do not take the CSE into account, then the SED of the star will appear fainter in the optical and, as consequence, colder than it really is. Groenewegen (2020a) also notes that photometric temperatures of Cepheids are found in average 200 K cooler than spectroscopic determination. This might be a hint for a common photometric biased caused by a CSE.

6.8. Evolutionary stage of Y Oph

As mentioned previously, Y Oph most likely undergoes the 3rd crossing of the instability strip. This identification follows from its rate of period change. However, Y Oph is the second Cepheid known after Polaris to exhibit a decreasing light curve amplitude (Fernie 1990; Fernie et al. 1995a), with a slope of about −1 mmag yr−1 similar to Polaris (Fernie et al. 1993). An examination of the V-band light curve over the past century suggests that its amplitude might have been around 0.7 mag at the beginning of the 20th century, compared to its current value of 0.5 mag. Therefore, the location close to the blue edge together with the decreasing light amplitude of Y Oph leads us to consider this star as being on its second crossing, leaving the instability strip. However, an in-depth statistical analysis of the light amplitude decline performed by Pop et al. (2010) has shown that drawing a firm conclusion on an amplitude change in Y Oph is not possible with the available data.

6.9. Absolute magnitude of Y Oph

Kovtyukh et al. (2010) empirically determined the absolute visual magnitude of Y Oph to be MV = −3.90 ± 0.15 mag at the pulsation phase ϕ = 0.454. This is puzzling, since such a magnitude would require a low mass of about 3 M and it is also discrepant with the Gaia distance. Their empirical relation is calibrated from FeII/FeI line depth ratios of F and G supergiants and is shown to be accurate for application of Cepheids. They applied their method from Cepheid spectra observed around maximum radius (ϕ ≈ 0.40) to mitigate dynamical effects. However, they scrutinized the application of this method along the pulsation cycle in the particular case of δ Cep. They find absolute magnitudes up to 0.8 mag fainter at phases between minimum and maximum light (ϕ ≈ 0.75 − 1.00) compared to expected values. Therefore, we think that systematic errors cannot be excluded for Y Oph as it demonstrates a different dynamical behavior than the bulk of Cepheids.

7. Conclusion

Y Oph is a long-period Cepheid known in the literature for its small light amplitude and low luminosity, among the other peculiarities discussed at the start of this paper. We carried out a hydrodynamical modeling with MESA-RSP in combination with a full set of observations to constrain the physical characteristics of this star. This comparison with the observations has allowed us to constrain the mass and the distance of the star. We have thus drawn the following conclusions:

  1. On the basis of the linear nonadiabatic analysis, we conclude that Y Oph is a fundamental-mode Cepheid.

  2. From the non-linear analysis, we find that the low radial velocity and light-curve amplitude of Y Oph is caused by the proximity of the star to the blue edge of the instability strip.

  3. We find that the Gaia DR2 and DR3 distances of Y Oph are in agreement with models of pulsational mass of about 7–8 M. These masses are also in agreement with the masses inferred by non-canonical evolutionary models, assuming a mild convective core overshoot. The luminosity is also consistent with PL relations calibrated in the Ks band.

  4. On the other hand, the BW distances determinations from the literature are discrepant with Gaia distances, while they are consistent with a pulsation mass of 5 M or less. Such a combination of mass and distance cannot be explained by evolutionary models. Moreover, the luminosity derived from pulsation models is much lower than that expected from the PL relation.

  5. We find that the distance of Y Oph derived by the BW method based on SBCR is biased if the impact of the CSE on the photometry is not taken into account. In addition, our result suggests that the p-factor of Y Oph might be close to the geometric limit of 1.5.

In conclusion, pulsation modeling with the MESA/RSP code is a valuable tool for providing constraints on the physical parameters of Cepheids. However, exploring the physics of the CSE is crucial to refine the photometric modeling of Cepheids and make it more realistic. Lastly, we stress that Y Oph is an important Cepheid with respect to improving our understanding of the physics behind the p-factor. Nevertheless, Y Oph is a long-period Cepheid with a reliable Gaia parallax that can be useful to calibrating the PL relation. The development of pulsation models with a higher grid resolution will be essential for a more precise determination of the p-factor and the physical parameters of this star. On the other hand, a thorough investigation of the evolutionary state of Y Oph will be important for drawing a firm conclusion about its nature.


Acknowledgments

We thank the referee for their helpful comments and suggestions. V.H., R.S., O.Z., R.S.R. are supported by the National Science Center, Poland, Sonata BIS project 2018/30/E/ST9/00598. This research made use of the SIMBAD and VIZIER databases at CDS, Strasbourg (France) and the electronic bibliography maintained by the NASA/ADS system. This research also made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration 2018). This research has made use of the Spanish Virtual Observatory (https://svo.cab.inta-csic.es) project funded by MCIN/AEI/10.13039/501100011033/ through grant PID2020-112949GB-I00.

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Appendix A: MESA/RSP inlist

A.1. MESA inlist

&star_job

    show_log_description_at_start = .false.

    create_RSP_model = .true.

    save_model_when_terminate = .true.
    save_model_filename = 'final.mod'

    initial_zfracs = 6

    ! History and profile columns
    history_columns_file = 'history_columns.list'

    set_initial_age = .true.
    initial_age = 0

    set_initial_model_number = .true.
    initial_model_number = 0

    set_initial_cumulative_energy_error = .true.
    new_cumulative_energy_error = 0d0

    pgstar_flag = .false.

/ ! end of star_job namelist

&eos
/ ! end of eos namelist

&kap
    Zbase = 0.0134
    kap_file_prefix = 'a09'
    kap_lowT_prefix = 'lowT_fa05_a09p'
    kap_CO_prefix = 'a09_co'
/ ! end of kap namelist

&controls
!max_model_number = 2 ->  to uncomment for LNA
! limit max_model_number as part of test_suite
     RSP_max_num_periods = 6000
! True convergence criteria defined in run_star_extras

! RSP controls

    RSP_do_check_omega = .true.

    ! Mass, Teff, luminosity and hydrogen abundance
    ! Set by the next script provided in A.2
    RSP_mass = mmmm
    RSP_Teff = tttt
    RSP_L    = llll
    RSP_X    = xxxx
    RSP_Z    = 0.0134

    !Uncomment for LNA
    !RSP_nmodes = 12 ! number of modes analyzed
    RSP_kick_vsurf_km_per_sec = 10d0
    RSP_fraction_1st_overtone = 0d0
    RSP_fraction_2nd_overtone = 0d0

! :: SET A OF CONVECTIVE PARAMETERS
    RSP_alfa   = 1.5d0
    RSP_alfac  = 1.0d0
    RSP_alfas  = 1.0d0
    RSP_alfad  = 1.0d0
    RSP_alfap  = 0.0d0
    RSP_alfat  = 0.0d0
    RSP_alfam  = 0.25d0
    RSP_gammar = 0.0d0

! controls for building the initial model
    RSP_nz = 150
    RSP_nz_outer = 40
    RSP_T_anchor = 11d3
    RSP_T_inner = 2d6
    RSP_target_steps_per_cycle = 600

! output controls
    terminal_show_age_units = 'days'
    trace_history_value_name(1) = 'rel_E_err'
    trace_history_value_name(2) = 'log_rel_run_E_err'
    photo_interval    = 1000
    profile_interval  = -1
    history_interval  = 1
    terminal_interval = 4000
/ ! end of controls namelist

A.2. Script

Below we provide a useful bash script that can be used or adapted for conveniently prepare and run several models for the previous inlist, with parameters defined in a text file.

file="parameters_list.dat"

# number of models to be computed
n=$(wc -l $file |awk '{print $1}')

for i in $(seq 1 $n);
do
    # l, t, m & x in a single variable
    arg=$(head -n $i $file |tail -n 1)

    # separate variables for l, t, m & z
    l=$(head -n $i $file |tail -n 1 | awk '{print $1}')
    t=$(head -n $i $file |tail -n 1 | awk '{print $2}')
    m=$(head -n $i $file |tail -n 1 | awk '{print $3}')
    x=$(head -n $i $file |tail -n 1 | awk '{print $4}')
    echo $l $t $m $x

    cp inlist_project_ltz inlist_project
    sed -i "s/llll/$l/g" inlist_project
    sed -i "s/tttt/$t/g" inlist_project
    sed -i "s/mmmm/$m/g" inlist_project
    sed -i "s/xxxx/$x/g" inlist_project
done

Appendix B: Results of the non-linear analysis

thumbnail Fig. B.1.

Best result of the non-linear analysis with RSP for 3 M.

thumbnail Fig. B.2.

Best result of the non-linear analysis with RSP for 4 M.

thumbnail Fig. B.3.

Best result of the non-linear analysis with RSP for 5 M.

thumbnail Fig. B.4.

Best result of the non-linear analysis with RSP for 6 M.

thumbnail Fig. B.5.

Best result of the non-linear analysis with RSP for 7 M.

All Tables

Table 1.

Spectroscopic metallicities gathered from the literature for Y Oph.

Table 2.

Models selected for non-linear computations, based on results of the LNA analysis for a linear period of P ≃ 17 days (see Fig. 1).

Table 3.

Best fits of the non-linear models.

Table 4.

Summary of Baade–Wesselink distances for Y Oph from the literature.

All Figures

thumbnail Fig. 1.

Results of the linear non-adiabatic (LNA) analysis with MESA-RSP. Positive and negative growth rates, γ for the fundamental mode are indicated in red and grey points, respectively. The black crosses are the selected models for non-linear analysis as summarized in Table 2. The vertical blue strip corresponds to the expected value of the mean effective temperature for Y Oph, i.e., Teff = 5800 ± 100 K (Luck 2018; Proxauf et al. 2018; da Silva et al. 2022). The fundamental blue edge from Anderson et al. (2016) is shown for comparison (see their Fig. 2: Z = 0.14, ωini = 0.5, mix between 2nd and 3rd crossing) with ΔL ± 500 L.

In the text
thumbnail Fig. 2.

Best result for the non-linear analysis with RSP/MESA for 8 M model. Uniform disk angular diameter, radial velocity curve, and effective temperature are displayed in (a), (b), and (c), respectively. The photometric panels indicate the apparent magnitudes in (d) V-band, (e) J-band, (f) H-band, (g) KS-band, (h) L-band, and (i) M-band. In the angular diameter and photometric panels, the thick black line and dashed grey lines are the RSP models, with and without the CSE models, respectively.

In the text
thumbnail Fig. 3.

O − C diagram of Y Oph. O − C calculations were performed by Csörnyei et al. (2022). The dashed line is a parabola fit over the entire range of data. The continuous line on the bottom plot is the linear fit of the O − C residuals over the observation period used in this paper.

In the text
thumbnail Fig. 4.

Fit of the parametric CSE model derived from Eq. (6) in the case of the best fit at 8 M. Errors bars represent the standard deviation of photometric measurements compared to the model. The grey region is the error on the magnitude obtained using the covariance matrix of the fitting result.

In the text
thumbnail Fig. 5.

Mean versus the mean effective temperature for models of different masses.

In the text
thumbnail Fig. 6.

Comparison of distance and absolute luminosity in the Ks band between the literature and our best results from non-linear models. (a) Comparison of distances obtained using variant of SBCR Baade-Wesselink methods based on (V − R) (Gieren 1988; Hindsley & Bell 1989; Gieren et al. 1993; V − K) (Storm et al. 2004, 2011; Barnes et al. 2005; Groenewegen 2008, 2013); interferometric Baade-Wesselink method (Mérand et al. 2007; Kervella et al. 2004a); distance obtained from the parallax measured by Gaia DR2 and DR3 (Gaia Collaboration 2022); and distance inferred from RSP models (see Table 3) as vertical strips for each stellar mass assuming uncertainty of ±15 pc. The average of both BW distances based on (V − K) color and their p-factor are indicated. (b) Comparison of KS-band absolute magnitude derived from different PL relations (Benedict et al. 2007; Fouqué et al. 2007; Gieren et al. 2018; Groenewegen 2018; Breuval et al. 2020, 2021; Trahin et al. 2021) established in the 2MASS filter (see Breuval et al. 2020), and the magnitude as we derived in the 2MASS filter from MESA-RSP non-linear models.

In the text
thumbnail Fig. 7.

Color excess E(B − V) in the direction of Y Oph versus the distance, from 3D extinction map (Lallement et al. 2014; Capitanio et al. 2017). The horizontal red strip indicates mean E(B − V) and standard deviation of the mean as compiled by Fernie et al. (1995a). Green and blue strips indicate the mean BW(V − K) distance and dGaia distance Gaia DR3 (Gaia Collaboration 2022), respectively.

In the text
thumbnail Fig. 8.

HR diagram comparing parameters of MESA-RSP models of Y Oph given in Table 2 (crosses) with evolutionary tracks computed with MESA for different stellar masses (MESA version r-21.12.1 used by Ziółkowska et al., in prep.). These models assume an overshooting parameter fov = 0.02. The vertical blue strip corresponds to the measured value of the mean effective temperature for Y Oph, i.e. Teff = 5800 ± 100 K.

In the text
thumbnail Fig. 9.

Comparison of SBCR derived for Y Oph with RSP along the pulsation cycle and empirical SBCR from Kervella et al. (2004b).

In the text
thumbnail Fig. 10.

Comparison of uniform disk angular diameter measurements in K-band from Mérand et al. (2007) with a combination of the Gaia DR3 distance and a p-factor of 1.5. The orange bar is a discrepant measurement compared to Gaia distance model (see Sect. 6.6).

In the text
thumbnail Fig. B.1.

Best result of the non-linear analysis with RSP for 3 M.

In the text
thumbnail Fig. B.2.

Best result of the non-linear analysis with RSP for 4 M.

In the text
thumbnail Fig. B.3.

Best result of the non-linear analysis with RSP for 5 M.

In the text
thumbnail Fig. B.4.

Best result of the non-linear analysis with RSP for 6 M.

In the text
thumbnail Fig. B.5.

Best result of the non-linear analysis with RSP for 7 M.

In the text

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